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[[546,236,4]] d =
n
546
k
236
d
4
kd²/n
6.916
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 3000 trials · survived 3000 trials · 2026-10-10
witness operator (support, 4 qubits)
[245, 306, 496, 503]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 3000 trials · survived 3000 trials · 2026-10-10
witness operator (support, 4 qubits)
[193, 251, 344, 405]
certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT
X: no logical < 4 exists; Z: no logical < 4 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7 · H_Z 7
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×546 (2,2)×3276 (3,2)×19656 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 546 (2,2): 3276 (3,2): 19656 (3,4): 5460
trapping sets H_Z (1,2)×546 (2,2)×3276 (3,2)×19656 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 546 (2,2): 3276 (3,2): 19656 (3,4): 5460

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Regular {7,7} hyperbolic surface code on a closed orientable surface. Quotient Q = Delta_7/Gamma with Q = PSL(2,13); the face-vertex swap is a projective involution J in PGL(2,13) imposing y = J x J-1, with ord(x) = ord(y) = 7 and ord(xy) = 2. Data qubits on edges = left cosets of the order-2 subgroup <xy>; Z checks on face boundaries = left cosets of <x>; X checks on vertex stars = left cosets of <y>. Checks commute because each face boundary meets each vertex an even number of times. n = |Q|/2 and k/n = (7-4)/7 + 2/n by Euler, giving k=236 at n=546. Regenerated from the algebra of arXiv:2610.10948 Sec. II by research/hyperbolic_psl.py; the matrices are recomputed, not copied, since that paper publishes its matrices on request only.
model human (claimed, not verified)
builds on arXiv:2610.10948
date 2026-10-10
notes Distance is a homological systole on each side, carried as explicit weight-4 witnesses, so d<=4 is witness-backed rather than certified exact. The paper's d=5 for this member is not reproduced here and is not claimed.
family topological (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[546,236,4]] [[546,236,4]], a self-dual PSL(2,13) {7,7} hyperbolic surface code: 236 logical qubits on 546 physical, a rate of 0.43 and five times the k of the incumbent [[546,46,8]] at identical blocklength.

Direction & hypothesis

weight-8 x unrestricted, family topological. Aimed at the k axis at fixed n: the board holds six entries at n=546 and the highest is [[546,46,8]] at weight 5. A weight-7 code at the same n with k=236 occupies a corner nothing else reaches.

What was searched

Generator-class enumeration for p=7, q=13, deduplicated on the check-support signature, giving more than 1500 classes all with n=546, k=236. Five classes were measured at 1500-3000 trials per side on a fresh seed; all five settled at d=4 on both sides and none reached the d=5 the paper reports for this member. The class filed is the first from the enumeration.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 1,000,000 (gf2_fast seed 101) | 4 | | 1,000,000 (gf2_fast seed 202) | 4 |

2 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,000,000 trials per side. The submitted witness reproduces it: X: d <= 4 (upper_bound), Z: d <= 4 (upper_bound).

Two fresh-seed rungs at 1,000,000 trials per side through the gf2_fast accelerator (seeds 101 and 202) both settle at weight 4, so the bound counts as settled. The claim is d<=4, NOT the paper's d=5 for this member: five of ~1500 generator classes were swept and all five returned 4, and the sweep was stopped before completion, so d=5 is neither claimed nor refuted.

Dead ends

  • A scan for a d=5 class was run over five measured classes at q=13 and then

    • over a wider budget, and returned d=4 every time; the scan was stopped rather than run to completion over the ~1500 classes, so this is an incomplete search, not a refutation of the paper's d=5.

  • At q=13 the class enumeration itself is the bottleneck: each candidate pair

    • re-derives the full order-|Q| closure, and the enumeration is what dominates the runtime before any distance search begins.

Tools

Model: human (opencode). The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py.

Reproduction

research/hyperbolic_psl.py --list 13 7 enumerates; --build 13 7 --class 0 writes HX/HZ as .npy. Matrices recomputed from the published construction, not copied (arXiv:2610.10948 offers its matrices on request only).

Parity checks

X-checks 156 (max weight 7) · Z-checks 156 (max weight 7)
H_X (156 checks, sparse supports)
[0, 5, 88, 187, 252, 298, 337] [1, 6, 89, 188, 254, 287, 326] [2, 7, 90, 189, 256, 289, 328] [3, 8, 78, 190, 258, 291, 330] [4, 9, 79, 191, 247, 293, 332] [5, 10, 80, 192, 249, 295, 334] [6, 11, 81, 193, 251, 297, 336] [7, 12, 82, 194, 253, 286, 325] [0, 8, 83, 182, 255, 288, 327] [1, 9, 84, 183, 257, 290, 329] [2, 10, 85, 184, 259, 292, 331] [3, 11, 86, 185, 248, 294, 333] [4, 12, 87, 186, 250, 296, 335] [13, 22, 226, 321, 351, 424, 474] [14, 23, 227, 322, 359, 419, 476] [15, 24, 228, 323, 354, 427, 478] [16, 25, 229, 324, 362, 422, 480] [13, 17, 230, 312, 357, 417, 469] [14, 18, 231, 313, 352, 425, 471] [15, 19, 232, 314, 360, 420, 473] [16, 20, 233, 315, 355, 428, 475] [17, 21, 221, 316, 363, 423, 477] [18, 22, 222, 317, 358, 418, 479] [19, 23, 223, 318, 353, 426, 468] [20, 24, 224, 319, 361, 421, 470] [21, 25, 225, 320, 356, 416, 472] [26, 32, 233, 272, 273, 312, 539] [27, 33, 225, 264, 278, 317, 541] [28, 34, 230, 269, 283, 322, 543] [29, 35, 222, 261, 275, 314, 545] [30, 36, 227, 266, 280, 319, 534] [31, 37, 232, 271, 285, 324, 536] [32, 38, 224, 263, 277, 316, 538] [26, 33, 229, 268, 282, 321, 540] [27, 34, 221, 260, 274, 313, 542] [28, 35, 226, 265, 279, 318, 544] [29, 36, 231, 270, 284, 323, 533] [30, 37, 223, 262, 276, 315, 535] [31, 38, 228, 267, 281, 320, 537] [39, 50, 438, 453, 496, 509, 530] [40, 51, 439, 454, 494, 507, 528] [39, 41, 440, 442, 505, 518, 526] [40, 42, 441, 443, 503, 516, 524] [41, 43, 429, 444, 501, 514, 522] [42, 44, 430, 445, 499, 512, 520] [43, 45, 431, 446, 497, 510, 531] [44, 46, 432, 447, 495, 508, 529] [45, 47, 433, 448, 506, 519, 527] [46, 48, 434, 449, 504, 517, 525] [47, 49, 435, 450, 502, 515, 523] [48, 50, 436, 451, 500, 513, 521] [49, 51, 437, 452, 498, 511, 532] [52, 53, 89, 104, 167, 182, 350] [53, 54, 84, 112, 162, 190, 348] [54, 55, 79, 107, 157, 185, 346] [55, 56, 87, 115, 165, 193, 344] [56, 57, 82, 110, 160, 188, 342] [57, 58, 90, 105, 168, 183, 340] [58, 59, 85, 113, 163, 191, 338] [59, 60, 80, 108, 158, 186, 349] [60, 61, 88, 116, 166, 194, 347] [61, 62, 83, 111, 161, 189, 345] [62, 63, 78, 106, 156, 184, 343] [63, 64, 86, 114, 164, 192, 341] [52, 64, 81, 109, 159, 187, 339] [65, 68, 101, 141, 180, 435, 442] [66, 69, 99, 139, 178, 440, 447] [67, 70, 97, 137, 176, 432, 452] [68, 71, 95, 135, 174, 437, 444] [69, 72, 93, 133, 172, 429, 449] [70, 73, 91, 131, 170, 434, 454] [71, 74, 102, 142, 181, 439, 446] [72, 75, 100, 140, 179, 431, 451] [73, 76, 98, 138, 177, 436, 443] [74, 77, 96, 136, 175, 441, 448] [65, 75, 94, 134, 173, 433, 453] [66, 76, 92, 132, 171, 438, 445] [67, 77, 103, 130, 169, 430, 450] [91, 169, 340, 366, 414, 467, 482] [92, 170, 343, 369, 404, 458, 486] [93, 171, 346, 372, 407, 462, 490] [94, 172, 349, 375, 410, 466, 481] [95, 173, 339, 365, 413, 457, 485] [96, 174, 342, 368, 403, 461, 489] [97, 175, 345, 371, 406, 465, 493] [98, 176, 348, 374, 409, 456, 484] [99, 177, 338, 364, 412, 460, 488] [100, 178, 341, 367, 415, 464, 492] [101, 179, 344, 370, 405, 455, 483] [102, 180, 347, 373, 408, 459, 487] [103, 181, 350, 376, 411, 463, 491] [117, 145, 195, 210, 337, 512, 525] [118, 146, 196, 211, 326, 517, 530] [119, 147, 197, 212, 328, 509, 522] [120, 148, 198, 213, 330, 514, 527] [121, 149, 199, 214, 332, 519, 532] [122, 150, 200, 215, 334, 511, 524] [123, 151, 201, 216, 336, 516, 529] [124, 152, 202, 217, 325, 508, 521] [125, 153, 203, 218, 327, 513, 526] [126, 154, 204, 219, 329, 518, 531] [127, 155, 205, 220, 331, 510, 523] [128, 143, 206, 208, 333, 515, 528] [129, 144, 207, 209, 335, 507, 520] [130, 207, 208, 359, 385, 392, 418] [131, 195, 209, 362, 388, 395, 421] [132, 196, 210, 352, 378, 398, 424] [133, 197, 211, 355, 381, 401, 427] [134, 198, 212, 358, 384, 391, 417] [135, 199, 213, 361, 387, 394, 420] [136, 200, 214, 351, 377, 397, 423] [137, 201, 215, 354, 380, 400, 426] [138, 202, 216, 357, 383, 390, 416] [139, 203, 217, 360, 386, 393, 419] [140, 204, 218, 363, 389, 396, 422] [141, 205, 219, 353, 379, 399, 425] [142, 206, 220, 356, 382, 402, 428] [104, 125, 143, 164, 236, 300, 476] [105, 126, 144, 165, 238, 302, 480] [106, 127, 145, 166, 240, 304, 471] [107, 128, 146, 167, 242, 306, 475] [108, 129, 147, 168, 244, 308, 479] [109, 117, 148, 156, 246, 310, 470] [110, 118, 149, 157, 235, 299, 474] [111, 119, 150, 158, 237, 301, 478] [112, 120, 151, 159, 239, 303, 469] [113, 121, 152, 160, 241, 305, 473] [114, 122, 153, 161, 243, 307, 477] [115, 123, 154, 162, 245, 309, 468] [116, 124, 155, 163, 234, 311, 472] [234, 264, 273, 303, 456, 488, 495] [235, 265, 274, 304, 458, 490, 504] [236, 266, 275, 305, 460, 492, 500] [237, 267, 276, 306, 462, 481, 496] [238, 268, 277, 307, 464, 483, 505] [239, 269, 278, 308, 466, 485, 501] [240, 270, 279, 309, 455, 487, 497] [241, 271, 280, 310, 457, 489, 506] [242, 272, 281, 311, 459, 491, 502] [243, 260, 282, 299, 461, 493, 498] [244, 261, 283, 300, 463, 482, 494] [245, 262, 284, 301, 465, 484, 503] [246, 263, 285, 302, 467, 486, 499] [247, 286, 368, 386, 394, 412, 534] [248, 287, 376, 381, 402, 407, 537] [249, 288, 371, 389, 397, 415, 540] [250, 289, 366, 384, 392, 410, 543] [251, 290, 374, 379, 400, 405, 533] [252, 291, 369, 387, 395, 413, 536] [253, 292, 364, 382, 390, 408, 539] [254, 293, 372, 377, 398, 403, 542] [255, 294, 367, 385, 393, 411, 545] [256, 295, 375, 380, 401, 406, 535] [257, 296, 370, 388, 396, 414, 538] [258, 297, 365, 383, 391, 409, 541] [259, 298, 373, 378, 399, 404, 544]
H_Z (156 checks, sparse supports)
[0, 88, 182, 347, 350, 459, 491] [1, 89, 183, 340, 350, 463, 482] [2, 90, 184, 340, 343, 467, 486] [3, 78, 185, 343, 346, 458, 490] [4, 79, 186, 346, 349, 462, 481] [5, 80, 187, 339, 349, 466, 485] [6, 81, 188, 339, 342, 457, 489] [7, 82, 189, 342, 345, 461, 493] [8, 83, 190, 345, 348, 465, 484] [9, 84, 191, 338, 348, 456, 488] [10, 85, 192, 338, 341, 460, 492] [11, 86, 193, 341, 344, 464, 483] [12, 87, 194, 344, 347, 455, 487] [13, 226, 312, 373, 408, 539, 544] [14, 227, 313, 368, 403, 534, 542] [15, 228, 314, 376, 411, 537, 545] [16, 229, 315, 371, 406, 535, 540] [17, 230, 316, 366, 414, 538, 543] [18, 231, 317, 374, 409, 533, 541] [19, 232, 318, 369, 404, 536, 544] [20, 233, 319, 364, 412, 534, 539] [21, 221, 320, 372, 407, 537, 542] [22, 222, 321, 367, 415, 540, 545] [23, 223, 322, 375, 410, 535, 543] [24, 224, 323, 370, 405, 533, 538] [25, 225, 324, 365, 413, 536, 541] [26, 118, 146, 233, 321, 474, 475] [27, 127, 155, 225, 313, 471, 472] [28, 123, 151, 230, 318, 468, 469] [29, 119, 147, 222, 323, 478, 479] [30, 128, 143, 227, 315, 475, 476] [31, 124, 152, 232, 320, 472, 473] [32, 120, 148, 224, 312, 469, 470] [33, 129, 144, 229, 317, 479, 480] [34, 125, 153, 221, 322, 476, 477] [35, 121, 149, 226, 314, 473, 474] [36, 117, 145, 231, 319, 470, 471] [37, 126, 154, 223, 324, 468, 480] [38, 122, 150, 228, 316, 477, 478] [39, 132, 141, 378, 399, 438, 442] [40, 138, 142, 382, 390, 439, 443] [41, 135, 139, 386, 394, 440, 444] [42, 132, 136, 377, 398, 441, 445] [43, 133, 142, 381, 402, 429, 446] [44, 130, 139, 385, 393, 430, 447] [45, 136, 140, 389, 397, 431, 448] [46, 133, 137, 380, 401, 432, 449] [47, 130, 134, 384, 392, 433, 450] [48, 131, 140, 388, 396, 434, 451] [49, 137, 141, 379, 400, 435, 452] [50, 134, 138, 383, 391, 436, 453] [51, 131, 135, 387, 395, 437, 454] [52, 89, 187, 196, 210, 326, 337] [53, 84, 182, 204, 218, 327, 329] [54, 79, 190, 199, 213, 330, 332] [55, 87, 185, 207, 208, 333, 335] [56, 82, 193, 202, 216, 325, 336] [57, 90, 188, 197, 211, 326, 328] [58, 85, 183, 205, 219, 329, 331] [59, 80, 191, 200, 214, 332, 334] [60, 88, 186, 195, 209, 335, 337] [61, 83, 194, 203, 217, 325, 327] [62, 78, 189, 198, 212, 328, 330] [63, 86, 184, 206, 220, 331, 333] [64, 81, 192, 201, 215, 334, 336] [65, 242, 306, 435, 453, 496, 502] [66, 238, 302, 440, 445, 499, 505] [67, 234, 311, 432, 450, 495, 502] [68, 243, 307, 437, 442, 498, 505] [69, 239, 303, 429, 447, 495, 501] [70, 235, 299, 434, 452, 498, 504] [71, 244, 308, 439, 444, 494, 501] [72, 240, 304, 431, 449, 497, 504] [73, 236, 300, 436, 454, 494, 500] [74, 245, 309, 441, 446, 497, 503] [75, 241, 305, 433, 451, 500, 506] [76, 237, 301, 438, 443, 496, 503] [77, 246, 310, 430, 448, 499, 506] [28, 91, 170, 265, 283, 458, 482] [31, 92, 171, 267, 285, 462, 486] [34, 93, 172, 269, 274, 466, 490] [37, 94, 173, 271, 276, 457, 481] [27, 95, 174, 260, 278, 461, 485] [30, 96, 175, 262, 280, 465, 489] [33, 97, 176, 264, 282, 456, 493] [36, 98, 177, 266, 284, 460, 484] [26, 99, 178, 268, 273, 464, 488] [29, 100, 179, 270, 275, 455, 492] [32, 101, 180, 272, 277, 459, 483] [35, 102, 181, 261, 279, 463, 487] [38, 103, 169, 263, 281, 467, 491] [52, 104, 159, 239, 269, 283, 300] [57, 105, 160, 241, 271, 285, 302] [62, 106, 161, 243, 260, 274, 304] [54, 107, 162, 245, 262, 276, 306] [59, 108, 163, 234, 264, 278, 308] [64, 109, 164, 236, 266, 280, 310] [56, 110, 165, 238, 268, 282, 299] [61, 111, 166, 240, 270, 284, 301] [53, 112, 167, 242, 272, 273, 303] [58, 113, 168, 244, 261, 275, 305] [63, 114, 156, 246, 263, 277, 307] [55, 115, 157, 235, 265, 279, 309] [60, 116, 158, 237, 267, 281, 311] [44, 109, 117, 151, 159, 512, 529] [48, 110, 118, 152, 160, 517, 521] [39, 111, 119, 153, 161, 509, 526] [43, 112, 120, 154, 162, 514, 531] [47, 113, 121, 155, 163, 519, 523] [51, 114, 122, 143, 164, 511, 528] [42, 115, 123, 144, 165, 516, 520] [46, 116, 124, 145, 166, 508, 525] [50, 104, 125, 146, 167, 513, 530] [41, 105, 126, 147, 168, 518, 522] [45, 106, 127, 148, 156, 510, 527] [49, 107, 128, 149, 157, 515, 532] [40, 108, 129, 150, 158, 507, 524] [8, 255, 291, 360, 387, 393, 420] [9, 257, 293, 363, 377, 396, 423] [10, 259, 295, 353, 380, 399, 426] [11, 248, 297, 356, 383, 402, 416] [12, 250, 286, 359, 386, 392, 419] [0, 252, 288, 362, 389, 395, 422] [1, 254, 290, 352, 379, 398, 425] [2, 256, 292, 355, 382, 401, 428] [3, 258, 294, 358, 385, 391, 418] [4, 247, 296, 361, 388, 394, 421] [5, 249, 298, 351, 378, 397, 424] [6, 251, 287, 354, 381, 400, 427] [7, 253, 289, 357, 384, 390, 417] [66, 99, 171, 247, 293, 372, 412] [69, 93, 178, 248, 294, 367, 407] [72, 100, 172, 249, 295, 375, 415] [75, 94, 179, 250, 296, 370, 410] [65, 101, 173, 251, 297, 365, 405] [68, 95, 180, 252, 298, 373, 413] [71, 102, 174, 253, 286, 368, 408] [74, 96, 181, 254, 287, 376, 403] [77, 103, 175, 255, 288, 371, 411] [67, 97, 169, 256, 289, 366, 406] [70, 91, 176, 257, 290, 374, 414] [73, 98, 170, 258, 291, 369, 409] [76, 92, 177, 259, 292, 364, 404] [17, 200, 216, 357, 423, 516, 524] [18, 207, 210, 352, 418, 512, 520] [19, 201, 217, 360, 426, 508, 529] [20, 195, 211, 355, 421, 517, 525] [21, 202, 218, 363, 416, 513, 521] [22, 196, 212, 358, 424, 509, 530] [23, 203, 219, 353, 419, 518, 526] [24, 197, 213, 361, 427, 514, 522] [25, 204, 220, 356, 422, 510, 531] [13, 198, 214, 351, 417, 519, 527] [14, 205, 208, 359, 425, 515, 523] [15, 199, 215, 354, 420, 511, 532] [16, 206, 209, 362, 428, 507, 528]
Code ID 546-236-4 · download JSON · raw on GitHub